Symmetric webs, Jones-Wenzl recursions and $q$-Howe duality

We define and study the category of symmetric ${\mathfrak{s}\mathfrak{l}}_2$-webs. This category is a combinatorial description of the category of all finite dimensional quantum ${\mathfrak{s}\mathfrak{l}}_2$-modules. Explicitly, we show that (the additive closure of) the symmetric ${\mathfrak{s}\mathfrak{l}}_2$-spider is (braided monoidally) equivalent to the latter. Our main tool is a quantum version of symmetric Howe duality. As a corollary of our construction, we provide new insight into Jones-Wenzl projectors and the colored Jones polynomials.